Problem with rear drum brakes

I am a total rookie when it comes to drum brakes, so now I wonder: How do I know that they are fitted correctly? Look at those washers on top mount for the brake shoes (where the two springs are attached). Should they really be there? Are they needed to push the brake shoes in place against the plate?

If I want to buy spare parts (spring, shoes, dust covers for the two adjust holes in the back of the plate, etc), where do I find them?

Generally the best illustrations are in the service manuals. This was said before, I'm just agreeing. But there were changes from year to year. So the best for your will be1963.

Next, especially for you, will be all the tips and advice Chrysler offered their techs.
Most of those films and pamphlets are available on-line in a few different formats.
There's one for 1963 here: Master Technician Service Conference Films - 1960 to 1963
and you can scroll down this list for more before and after 63:
Master Technician Service Conference - Chrysler's Training for Mechanics

There's one in '67 about brake hydraulics and another in '69 about drum brakes. But be aware that in '67 the tandem brake master cylinders were introduced; and in '69 uses a new style of self-adjuster. That said, there's useful service tips and diagnostic concepts in both.

You should be able to get parts but it may require some hunting. Depends on how many other vehicles it was used on, and whether it was correctly entered into a catalog.